15.12: Bases Ortonormales en Espacios Reales y Complejos
- Page ID
- 86577
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Notación
Transpose operador\(A^T\) voltea la matriz a través de su diagonal.
\ [\ begin {array} {l}
A=\ left (\ begin {array} {ll}
a_ {1,1} & a_ {1,2}\
a_ {2,1} & a_ {2,2}
\ end {array}\ derecha)\\
A^ {T} =\ left (\ begin {array} {ll}
a_ {1,1} & a_ {2,1}\
a_ {1,2} & a_ _ {2,2}
\ end {array}\ right)
\ end {array}\ nonumber\]
Columna\(i\) de\(A\) es fila\(i\) de\(A^T\)
Recall, producto interno
\ [\ boldsymbol {x} =\ left (\ begin {array} {c}
x_ {0}\\
x_ {1}\\
\ vdots\\
x_ {n-1}
\ end {array}\ derecha)\ nonumber\]
\ [\ boldsymbol {y} =\ left (\ begin {array} {c}
y_ {0}\\
y_ {1}\\
\ vdots\\
y_ {n-1}
\ end {array}\ derecha)\ nonumber\]
\ [\ negridsymbol {x} ^ {T}\ negridsymbol {y} =\ left (\ begin {array} {cccc}
x_ {0} & x_ {1} &\ ldots & x_ {n-1}
\ end {array}\ derecha)\ left (\ begin {array} {c}
y_ {0}\
y_ {1}\\
\ vdots\
y_ _ {n-1}
\ end {array}\ derecha) =\ suma_ { i}\ negridsymbol {x} _ {i}\ negridsymbol {y} _ _ {i} =\ langle\ negridsymbol {y},\ negridsymbol {x}\ rangle\ nonumber\]
encendido\(\mathbb{R}^n\).
Transposición hermitiana\(A^H\), transposición y conjugación
\ [\ begin {array} {c}
A^ {\ mathrm {H}} =\ overline {A^ {T}}\\ langle
\ negridsymbol {y},\ negridsymbol {x}\ rangle=\ negridsymbol {x} ^ {\ mathrm {H}}\ negridsymbol {y} =\ sum_ {i}\ negridsymbol {x} _ i}\ bar {\ boldsymbol {y}} _ {i}
\ end {array}\ nonumber\]
encendido\(\mathbb{C}^n\).
Ahora, dejemos\(\left\{b_{0}, b_{1}, \dots, b_{n-1}\right\}\) ser una base ortonormal para\(\mathbb{C}^n\)
\ [\ begin {array} {c}
i=\ {0,1,\ ldots, n-1\}\ izquierda\ langle b_ {i}, b_ {i}\ derecha\ rangle=1\
\ izquierda (i\ neq j,\ izquierda\ langle b_ {i}, b_ {i}\ derecha\ rangle=b_ {j} ^ {H} b_ {i} =0\ derecha)
\ end {array}\ nonumber\]
Matriz de base:
\ [B=\ left (\ begin {array} {cccc}
\ vdots &\ vdots &\ vdots\\
b_ {0} & b_ {1} &\ dots & b_ {n-1}\\\ vdots &
\ vdots &\ vdots &\ vdots
\ end {array}\ right)\ nonumber\]
Ahora,
\ [B^ {\ mathrm {H}} B=\ left (\ begin {array} {ccc}
\ cdots & b_ {0} ^ {\ mathrm {H}} &\ cdots\
\ cdots & b_ {1} ^ {\ mathrm {H}} &\ cdots\\
&\ vdots\
\ cdots & b_ {n-1} ^ {\ mathrm {H}} &\ ldots
\ end {array}\ derecha)\ left (\ begin {array } {cccc}
\ vdots &\ vdots &\ vdots\\
b_ {0} & b_ {1} &\ cdots & b_ {n-1}\\
\ vdots &\ vdots & &\ vdots
\ end {array}\ derecha) =\ left (\ begin {array} {cccc}
b_ {0} ^ {\ mathrm {H}} b_ {0}} & b_ {0} ^ {\ mathrm {H}} b_ {1} &\ ldots & b_ {0} ^ {\ mathrm {H}} b_ {n-1}\\
b_ {1} ^ {\ mathrm {H}} b_ {0} & b_ {1} ^ {\ mathrm {H}} b_ {1} &\ ldots & b_ {1} ^ {\ mathrm {H}} b_ {n-1}\
\ vdots & &\\
b_ {n-1} ^ {\ mathrm {H}} b_ {0} & b_ {n-1} ^ {\ mathrm {H}} b_ {1} &\ ldots & b_ {n-1} ^ {\ mathrm {H}} b_ {n-1}
\ end {array}\ derecha)\ nonumber\]
Para base ortonormal con matriz de base\(B\)
\[B^H=B^{-1} \nonumber \]
(\(B^{T}=B^{-1} \text {in } \mathbb{R}^{n}\)in\(\mathbb{R}^n\))\(B^H\) es fácil de calcular mientras que\(B^{-1}\) es difícil de calcular.
Entonces, para encontrar\(\left\{\alpha_{0}, \alpha_{1}, \ldots, \alpha_{n-1}\right\}\) tal que
\[\boldsymbol{x}=\sum_{i} \alpha_{i} b_{i} \nonumber \]
Calcular
\[\left(\boldsymbol{\alpha}=B^{-1} \boldsymbol{x}\right) \Rightarrow\left(\boldsymbol{\alpha}=B^{H} \boldsymbol{x}\right) \nonumber \]
Utilizando una base ortonormal nos libramos de la operación inversa.